Stability Analyses Of Staged-progression Epidemic Models With Nonlinear Incidence Rates And Distributed Delay

Infective diseases dynamics is an important part of biomathematics.It is a combination of epidemiology and ecological mathematics.Under the quantitative and qualitative analyses of the dynamic behaviors of epidemic models and the numerical simulations,the development process and law of the disease could be revealed,and the trend of infective diseases could be predicted,and then the best strategy for prevention could be founded.So far there are many scholars have proposed and paid efforts on the studies of various types of epidemic models with actual cases.This dissertation studies the single and multi-group staged-progression deterministic and stochastic epidemic models with distributed delays and nonlinear incidence rates.We obtain the sufficient conditions for the global asymptotic stability of the equilibrium by building the Lyapunov Functional and Lyapunov-Lasalle invariant principle.Also,by the introduction of stochastic perturbations,we establish a corresponding stochastic model and derive the sufficient conditions for the asympototic stability of the model.Finally,we complement our results by means of providing numerical simulations.This dissertation is divided into four chapters:Chapter 1 introduces the background and biological significance of epidemic models.On the basis of this,we thereby introduce the theoretical knowledge of staged-progression epidemic model.Chapter 2 gives our deterministic staged-progression epidemic model.We then prove the boundedness and positiveness of trajectories,and then discuss the existence of the equilibria together with their stability.Also,by the introduction of stochastic perturbations,we prove the respective stochastic asymptotic stability of the disease-free equilibrium and the endemic equilibrium.Chapter 3 introduces a multigroup deterministic staged-progression epidemic model with abstract nonlinear incidence and a general distributed time delay in which individuals may experience disease relapse.We thereby prove the boundedness and positiveness of trajectories,and then discuss the existence of the equilibria together with their stability.Also,by introducing stochastic perturbations of white noise,we prove the stochastic asymptotic stability of equilibrium.Finally,we complement our results by means of providing numerical simulations.Finally,we summarize the whole context of the dissertation and future research directions are discussed in Chapter 4.